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Specific hypotheses
Description 
Fundamental models are based on specific hypotheses that enable the development of models grounded in well-understood mechanisms. These models rely on established variables, constants, and equations, which typically require minimal parameter adjustment. This characteristic streamlines the modeling process and facilitates the validation procedures, ensuring that the hypothesis accurately explains the mechanisms involved.
This approach aids in creating robust and precise models that reflect the behavior of the studied system, supported by a solid theoretical framework. This enhances predictive capabilities and maintains consistency with empirical observations.
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Example of a specific hypothesis
Description
To illustrate how a hypothesis can structure a model and derive equations that accurately describe the behavior of a system, lets consider the case of an ideal gas. Under this approach, we establish the following fundamental hypotheses:
• The gas is composed of atoms or molecules that do not interact with each other.
If the gas is contained within a volume $V$, we can make the following additional assumptions:
• The particles are uniformly distributed throughout the volume, with no areas of higher or lower density.
• The motion of the particles is isotropic, meaning it is equally probable in all directions.
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Development of the model example
Description
The hypothesis establishes that the ideal gas:
• Is composed of particles.
• Has a uniform distribution.
• Moves isotropically.
• Exerts pressure through particle collisions with the walls.
Based on this last hypothesis, along with the previous ones, it can be concluded that the pressure $p$ is proportional to the number of collisions, which in turn is proportional to the concentration $c$:
$p \propto c$
Since concentration is inversely proportional to the volume $V$:
$c \propto \displaystyle\frac{1}{V}$
If the volume doubles, the concentration is halved, and so is the pressure. In other words, the pressure is inversely proportional to the volume:
$p \propto \displaystyle\frac{1}{V}$
This corresponds to Boyle's law:
$pV=constante$
In this way, Boyle's law is established directly from the hypotheses without the need to assume more complex mechanisms.
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A more sophisticated model
Description
If the mass of a particle is $m$ and it moves with an average velocity $v$ in a certain direction, the momentum it will transfer to the wall in an elastic collision is $2mv$.
Since the velocity is $v$, the distance traveled in a time interval $\Delta t$ is $v\Delta t$. Therefore, in a volume defined by an edge of length $v\Delta t$ and a surface area $S$, there is a total of $c v \Delta t S$ particles, where $c$ is the concentration. Of these particles, one-sixth will collide with the surface $S$ during the time $\Delta t$.
Thus, the total momentum transferred to the surface $\Delta p$ in the time $\Delta t$ is:
$2mv \displaystyle\frac{1}{6} c v \Delta t S$
Since force is the change in momentum per unit of time, we have:
$F=\displaystyle\frac{\Delta p}{\Delta t}=2mv \displaystyle\frac{1}{6} c v S$
and the resulting pressure is:
$pV = \displaystyle\frac{1}{3} N m v^2=constante$
Given that the concentration $c$ is equal to the number of particles $N$ divided by the volume $V$, the equation becomes Boyle's law:
$pV = \displaystyle\frac{1}{3} N m v^2=constante$
This more detailed model allows us to derive Boyle's law by considering the movement of particles and their elastic collisions with the walls.
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